1/36

HLbL contribution to the muon g − 2 on thelattice: overall strategy

Thomas Blum

UConn/RBRC

Norman Christ

Columbia

Masashi Hayakawa

Nagoya

Taku Izubuchi

BNL/RBRC

Luchang Jin

UConn/RBRC

Chulwoo Jung

BNL

Christoph Lehner

BNL

Cheng Tu

UConn

and the RBC/UKQCD collaborations

Mar 12, 2018

University of Connecticut

Muon g-2 Theory Initiative HLbL Working Group Workshop

Outline 2/36

• Introduction

• Lattice method

• HLbL on lattice

• Disconnected diagrams

• Finite volume effects

• π0 contribution

• Summary and Future plans

Schwinger’s term 3/36

Figure 1. The headstone of Julian Schwinger at Mt Auburn Cemetery in Cambridge, MA.

First measurement of muon magnetic moment 4/36

Current g − 2 experiment scheme 5/36

π+

Storage

Ring

Protons

from AGS

Target

Pions

p=3.1 GeV

π+→ µ

+νµ

Inflector

Polarized Muons

Injection Point

Kicker

Modules

Injection Orbit

Storage Ring Orbit

νµ µ+

⇒ ⇐ spinmomentum

In Pion Rest Frame

“Forward” Decay Muons

are highly polarized

Figure 2. The schematics of muon injection and storage in the g−2 ring. Phys. Rept. 477, 1 (2009).

ωc =eB

mµ γ(1)

ωs =eB

mµ γ+ aµ

eB

mµ(2)

γ = 1/ 1− v2√

≈ 29.3 (3)

List of experiments 6/36

World Average dominated by BNL

aµ = (11659208.9±6.3)× 10−10 (4)

In comparison, for electron

ae = (11596521.8073±0.0028)× 10−10 (5)

Future Fermilab E989 (0.14 ppm) 7/36

Figure 3. 1000 Piece Jigsaw Puzzle - Magnetic Moment. $18.00 from http://eddata.fnal.gov/ seems

not available anymore.

Almost 4 times more accurate then the previous experiment.

J-PARC E34 also plans to measure muon g−2 with similar precision.

5-loops QED calculation 8/36

Aoyama et al. ’12

Weak interaction contribution 9/36

Leading weak contribution. a= 38.87, b=−19.39, c= 0.00 [in units 10−10]

Value±Error ReferenceQED incl. 5-loops 11658471.8853± 0.0036 Aoyama, et al, 2012

Weak incl. 2-loops 15.36± 0.10 Gnendiger et al, 2013

Table 1. Standard model theory, QED and Weak interaction. [in units 10−10]

Hadronic contribution: HVP 10/36

q = p′ − p, ν

p p′

HVP: hadronic vaccum polarization

Leading order

• HVP LO: Errors are being reduced quite a bit. With lattice calculation and e+e− →hadrons experiments data combined, We obtained 692.5± 2.7. arXiv:1801.07224.

• HVP NLO: −9.84± 0.07, HLMNT11. arXiv:1105.3149

Hadronic contribution: HLbL 11/36

q = p′ − p, ν

p p′

q = p′ − p, ν

p p′

(L) HVP (R) HLbL

• HLbL: hadronic light-by-light scattering.

The following results are obtained with combinations of models on different process.

10.5± 2.6 PdRV09, Glasgow consensus, arXiv:0901.0306.

10.5± 4.9 same as above, but errors added linearly instead of in quadrature.

11.6± 3.9 JN09, arXiv:0902.3360.

10.3± 2.9 FJ17, arXiv:1705.00263.

• Dispersive approach.

• Lattice approach.

HLbL models 12/36

q = p′ − p, ν

p p′

Various contributions to aµHLbL× 1010

PdRV09 JN09 FJ17(Glasgow consensus)

π0, η, η ′ 11.4± 1.3 9.9± 1.6 9.5± 1.2

π,K loops −1.9± 1.9 −1.9± 1.3 −2.0± 0.5axial-vector 1.5± 1.0 2.2± 0.5 0.8± 0.3

scalar −0.7± 0.7 −0.7± 0.2 −0.6± 0.1quark loops 0.2 (charm) 2.1± 0.3 2.2± 0.4

tensor - - 0.1± 0.0NLO - - 0.3± 0.2

Total 10.5± 4.9 11.6± 3.9 10.3± 2.910.5± 2.6 (quadrature)

BNL E821 (0.54 ppm) and standard model prediction 13/36

q = p′ − p, ν

p p′

q = p′ − p, ν

p p′

(L) HVP (R) HLbL

aµ× 1010

HVP (e+e−→ hadrons) 692.5± 2.7 RBC/UKQCD and FJ17 combinedHadronic Light by Light 10.3± 2.9 FJ17

Standard Model 11659180.2± 3.9Experiment (0.54 ppm) 11659208.9± 6.3 E821, The g − 2 Collab. 2006Difference (Exp− SM) 28.7± 7.4

Table 2. Standard model theory and experiment comparison

There is 3.9 standard deviations!

Outline 14/36

• Introduction

• Lattice method

• HLbL on lattice

• Disconnected diagrams

• Finite volume effects

• π0 contribution

• Summary and Future plans

Lattice QCD 15/36

The QCD partition function in Euclidean space time:

Z =

∫

[DUµ]e−SG[U ]det(D[ml, U ])2 det(D[ms, U ]) (6)

(Left) 19× 19 Go board (Middle) 48× 48 (Right) 64× 64

The configuration is stored in position space. The reason is that the action is local in positionspace. Working in position makes the calculation simpler.

This is in contrast to analytical perturbative calculation, where interaction only happensoccasionally. So it is advantagous to work in momentum space, where the propagator can bediagonalized.

zMobius + Multigrid Lanczos + AMA 16/36

• We are using Domain wall fermion (DWF) in all our lattice calculations for HLbL. DWFrespects Chiral symmetry, which helps systematically eliminating the O(a) discretizationerror. The remaing discretization error are in general quite small. The fifth dimension isneeded in order to fullfil the Chiral symmetry. This results numerical cost proportion tothe length in the fifth dimension, Ls, and large Ls is needed to reach the Chiral limit.

• Mobius DWF allows us to use a smaller value for Ls and having almost the same Chiralproperty. For 48I, we use Ls= 24 to mimic the original Ls= 48 DWF.

• The zMobius formulation allows us to obtain a very good approximate of the original(M)DWF propagator with a significantly reduced Ls. For 48I, we further reduce the Ls

from 24 to 10. PoS LATTICE2015, 019 (2016)

• Multigrid Lanczos algorithm help us efficiently generate the low modes of the DWFoperator which accelerate the inversion roughly by a factor of 20 for light quarks.arXiv:1710.06884

• All-mode-averaging (AMA) allows us to perform the inversion with much less iterationsmost of the time, and only compute the small correction term for a small portion of theentire calculation. This can bring an addition factor of 5 speed up. Phys. Rev. D 91,no. 11, 114511 (2015)

• We use highly optimized DWF Dirac operator inverter from the BFM and Grid to performthe inversion. https://github.com/paboyle

Outline 17/36

• Introduction

• Lattice method

• HLbL on lattice

• Disconnected diagrams

• Finite volume effects

• π0 contribution

• Summary and Future plans

Hadronic light by light diagram on lattice 18/36

• This subject is started by T. Blum, S. Chowdhury, M. Hayakawa, T. Izubuchi more than7 years ago. hep-lat/0509016, Phys.Rev.Lett. 114 (2015) no.1, 012001.

• A series of improvements in methodology is made later. We computed the connecteddiagram of HLbL with 171 MeV pion mass. Phys.Rev. D93 (2016) no.1, 014503.

• Mainz group independently come up with a similar method to compute HLbL. PoSLATTICE2015 (2016) 109.

• With the improved methods, we calculated HLbL using the physical pion mass, 483 ×96, ensemble. And for the first time, we computed the leading disconnected diagramscontribution. Phys.Rev.Lett. 118 (2017) no.2, 022005.

• Mainz group announces the significant progress on the method to reduce the finite volumeeffects by treating the QED part of the HLbL diagram semi-analyticly in infinite volume.Part of the results are given in PoS LATTICE2016 (2016) 164.

• Encouraged by Mainz’s success, we use a different approach to compute the QED partof the HLbL in infinite volume. Based the results, we exploit a way to furthur reduce thelattice discretization error and finite volume error. Phys.Rev. D96 (2017), 034515.

RBC’s version of the history on this subject.

QED configuration v.s. Point source propagator 19/36

xsrc xsnk

Bm2

σ′ (y′)

z′, κ′

Am1

ρ′ (x′)

xop, ν

z, κ

Bm2

σ(y) Am1

ρ(x)

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

• Left: the two photon propagators are represented by ensemble average of QED configu-rations.

• Right: the two photon propagators are represented by random sampling of point sourcelocations.

• Goal: Keep the signal unchanged, reduce the noise as much as possible.

• From another point of view, the question becomes: What is best way to evalute the 4points correlation function on the lattice, which allows efficient extraction of the HLbLcontribution to muon g− 2.

Point source photon method 20/36

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

FνC(q~;x, y, z, xop)= (−ie)6 Gρ,σ,κ(q~;x, y, z)Hρ,σ,κ,ν

C (x, y, z, xop) (7)

i4Hρ,σ,κ,νC (x, y, z, xop) (8)

=∑

q=u,d,s

(eq/e)4

6

⟨

tr

[

−iγρSq(x, z)iγκSq(z, y)iγσSq(y, xop)iγνSq(xop, x)

]⟩

QCD

+ other 5 permutations

i3 Gρ,σ,κ(q~;x, y, z) (9)

= emµ

2+q~2/4√

(tsnk−tsrc)∑

x′,y ′,z ′

Gρ,ρ′(x, x′)Gσ,σ ′(y, y ′)Gκ,κ′(z, z ′)

×∑

x~ snk,x~ src

e−iq~/2·(x~ snk+x~ src)Sµ(xsnk, x′)iγρ′Sµ(x

′, z ′)iγκ′Sµ(z′, y ′)iγσ ′Sµ(y

′, xsrc)

+ other 5 permutations

Magentic moment - QFT 21/36

q = p′ − p, ν

p p′

q = p′ − p, µ

p p′

Figure 4. (L) Muon Vertex Function Diagram (R) Schwinger Term Diagram.

⟨

p~ ′, s′∣

∣jν(

x~ op=0~)∣

∣p~ , s⟩

=

⟨

p~ ′, s′∣

∣

∣

∣

∣

∑

f

qf ψf(x~ op=0) γνψf(x~ op=0)

∣

∣

∣

∣

∣

p~ , s

⟩

= −e us′(p~ ′)

[

F1(q2)γν + i

F2(q2)

4m[γν , γρ]qρ

]

us(p~) (10)

µ~ = −g e

2ms~ =−(F1(0)+F2(0))

e

ms~ (11)

F1(0) = 1 (12)

F2(0) =g− 2

2≡ a (13)

Magentic moment - E&M 22/36

Classicaly, magnetic moment is simply

µ~ =

∫

1

2x~ × j~ d3x (14)

• This formula is not correct in Quantum Mechanics, because the magnetic moment resultfrom the spin is not included.

• In Quantum Field Thoery, Dirac equation automatically predict fermion spin, so the naiveequation is correct again!

〈µ~ 〉 =

⟨

ψ

∣

∣

∣

∣

∫

1

2x~ op× i j~(x~ op) d

3xop

∣

∣

∣

∣

ψ

⟩

(15)

• i j~(x~ op) is the conventional Minkovski spatial current, because of our γ matrix convention.

• The right hand generate the total magnetic moment for the entire system, includingmagnetic moment from spin.

• Above formula applies to normalizable state with zero total current. Not practical onlattice because it need extremely large volume to evaluate.

L ≫ ∆xop∼ 1/∆p (16)

Magentic moment - moment method 23/36

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

F2(0)

mus′

(

0~)Σ~

2us(

0~)

=∑

r

[

∑

z,xop

1

2x~ op× us′

(

0~)

iF~ C(

0~ ;x=−r2, y=+

r

2, z, xop

)

us(

0~)

]

• The initial and final muon states are plane waves instead of properly normalized states.

• Recall the definition for Fµ, we sum all the internal points over the entire space timeexcept we fix x+ y=0.

• The time coordinate of the current, (xop)0 is integrated instead of being held fixed.

These features allow us to perform the lattice simulation efficiently in finite volume.

Note that we use the average of the two sampled points,1

2(x + y), as the reference point

(origin) for the moment method.

Magentic moment - moment method v2 24/36

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κ

y, σ x, ρ

F2(0)

mus′

(

0~)Σ~

2us(

0~)

=∑

r

[

∑

z,xop

1

2x~ op× us′

(

0~)

iF~ C(

0~ ;x=−r− z

2, y=

r− z

2, z, xop

)

us(

0~)

]

(17)

Instead of using the average of the two sampled points,1

2(x + y), as the reference point

(origin) for the moment method, we may also use the average of all three points1

3(x+ y+ z)

as the reference point (origin). The formula changes slightly and has the above form.

Then, all the three diagrams shown above become equivalent. We can then only evaluate onlythe first diagram with two point source propagators and no sequential propagators. Phys.Rev.D93 (2016) no.1, 014503

We plan to test this method with the infinite volume QED formulation.

Outline 25/36

• Introduction

• Lattice method

• HLbL on lattice

• Disconnected diagrams

• Finite volume effects

• π0 contribution

• Summary and Future plans

Disconnected diagrams 26/36

• One diagram (the biggest diagram below) do not vanish even in the SU(3) limit.

• We extend the method and computed this leading disconnected diagram as well.

xsrc xsnkz′, κ′ y′, σ′ x′, ρ′

xop, ν

z, κ y, σ x, ρ

xsrc xsnky′, σ′ x′, ρ′ z′, κ′

xop, ν

z, κy, σ x, ρ

xsrc xsnky′, σ′ z′, κ′ x′, ρ′

xop, ν

z, κy, σ x, ρ

xsrc xsnkz′, κ′ y′, σ′ x′, ρ′

xop, ν

z, κ y, σ x, ρ

xsrc xsnky′, σ′ x′, ρ′ z′, κ′

xop, ν

z, κy, σ x, ρ

xsrc xsnkz′, κ′

y′, σ′ x′, ρ′

xop, ν

z, κ y, σ x, ρ

• Permutations of the three internal photons are not shown.

• There should be gluons exchange between and within the quark loops, but are not drawn.

• We need to make sure that the loops are connected by gluons by “vacuum” subtraction.So the diagrams are 1-particle irreducible.

Disconnected formula 27/36

xsrc xsnkz′, κ′ y′, σ′ x′, ρ′

xop, ν

z, κ y, σ x, ρ

• Currently we have only calculated the leading disconnected diagram.

• Point z is used as the reference point (origin) for the moment method.

• We can use two point source photons at y and z, which are chosen randomly. The pointsxop and x are summed over exactly on lattice.

• Only point source quark propagators are needed. We computeM point source propagatorsand all M2 combinations of them are used to perform the stochastic sum over r= z− y.

Disconnected formula 28/36

xsrc xsnkz′, κ′ y′, σ′ x′, ρ′

xop, ν

z, κ y, σ x, ρ

F2dHLbL(0)

m

(σs′,s)i2

=∑

r,x

∑

xop

1

2ǫi,j ,k(xop)j · i us′

(

0~)

FkD(x, y= r, z=0, xop)us

(

0~)

(18)

FνD(x, y, z, xop) = (−ie)6 Gρ,σ,κ(x, y, z)Hρ,σ,κ,ν

D (x, y, z, xop) (19)

Hρ,σ,κ,νD (x, y, z, xop) =

⟨

1

2Πν,κ(xop, z) [Πρ,σ(x, y)−Πρ,σ

avg(x− y)]

⟩

QCD

(20)

Πρ,σ(x, y) = −∑

q

(eq/e)2Tr[γρSq(x, y) γσSq(y, x)]. (21)

Outline 29/36

• Introduction

• Lattice method

• HLbL on lattice

• Disconnected diagrams

• Finite volume effects

• π0 contribution

• Summary and Future plans

Infinite volume QED formulation 30/36

FνC(x, y, z, xop) = (−ie)6 Gρ,σ,κ(x, y, z)Hρ,σ,κ,ν(x, y, z, xop)

The QED part, Gρ,σ,κ(x, y, z) can be evaluated in infinite volume QED box.

The QCD part, Hρ,σ,κ,ν(x, y, z, xop) can be evaluated in a finite volume QCD box.

QCD Box

QED Box

z′

y′

x′

y

xz

xop

i3 Gρ,σ,κ(x, y, z) = Gρ,σ,κ(x, y, z)+Gσ,κ,ρ(y, z, x)+ other 4 permutations. (22)

Gρ,σ,κ(x, y, z) = emµ(tsnk−tsrc)∑

x′,y ′,z ′

Gρ,ρ′(x, x′)Gσ,σ ′(y, y ′)Gκ,κ′(z, z ′) (23)

×∑

x~ snk,x~ src

Sµ(xsnk, x′)iγρ′Sµ(x

′, y ′)iγσ ′Sµ(y′, z ′)iγκ′Sµ(z

′, xsrc)

QCD finite volume effects 31/36

Phys. Rev. D 93, 074505(2016)

48I: 483× 96, 5.5fm box 64I: 643× 128, 5.5fm box

24D: 243× 64, 4.8fm box 32D: 323× 64, 6.4fm box 48D: 483× 64, 9.6fm box

32Dfine: 323× 64, 4.8fm box

Outline 32/36

• Introduction

• Lattice method

• HLbL on lattice

• Disconnected diagrams

• Finite volume effects

• π0 contribution

• Summary and Future plans

π0 contribution 33/36

We can separate the HLbL contribution into the short distance part:

F2short(0)

mus′

(

0~)Σi

2us(

0~)

=1

V

∑

x,y,z

|x−y|,|y−z |,|x−z |6Lcut

∑

xop

1

2ǫi,j,k(xop)j iHρ,σ,κ,k(x, y, z, xop)

× us′

(

0~)

(−ie)6 Gρ,σ,κ(x, y, z) us(

0~)

And the remaining long distance part, where at least two vector currents are separated by adistance longer than Lcut.

The short distance part can be evaluted on the lattice with relatively small finite volume error.For the long distance part, the hadronic 4 point function can be approximated by two majorcontribution: π0 exchange and π± loop. We will start with the π0 exchange diagram:

Hρ,σ,κ,ν(x, y, z, xop) ≈∑

u,v

Fρ,σ(x, y, u)Gπ0(u, v)Fκ,ν(z, xop, u)+ permuations (24)

Outline 34/36

• Introduction

• Lattice method

• HLbL on lattice

• Disconnected diagrams

• Finite volume effects

• π0 contribution

• Summary and future plans

Summary and future plans 35/36

In a finite (5.5fm)3 box with inverse lattice spacing 1/a= 1.73 GeV, we obtained aµHLbL ×

1010= 5.35± 1.35, except for some subleading disconnected diagrams.

• Continuum limit: Finish up the finite volume QED 64I calculation to extrapolate awaythe discretization error for the 48I results.

• Connected infinite QED volume: QCD part can reuse the previously saved data. Pre-liminary results are available now.

• Leading disconnected infinite QED volume: Still, we plan to reuse the previously saveddata for the QCD part. We are exploring the method to suppress the statistics noise.

• Sub-leading disconnected diagrams: These diagrams vanishes in the flavor SU(3) limit.We plan to calculate these diagrams in the future.

• Large physical QCD volume: Study the QCD finite volume effects with the 24ID, 32ID,and 48ID ensemble. The 48I,64I and the 32IDfine ensembles can also help understand thediscretization effects with these coarse ensembles.

• Neutral pion contribution: Study the contribution from the π0 exchange at long dis-tance. This could help estimate or reduce the QCD finite volume error of the direct latticecalculation.

• Final goal is reaching 10% accuracy to compare with the new experiments.

Thank You 36/36

Thank You!

A New Hope 37/36

q = p′ − p, ν

p p′

aµ× 1010 ReferenceExperiment 11659208.9± 6.3 E821, g− 2 Collab. 2006

Standard Model 11659180.2± 3.9 Particle Data Group, 2014??? 28.7± 7.4

Future is hard to predict, let’s think of something similar in the history.

Precession of the perihelion of Mercury (in unit of arcsec/Julian century)

Precession ReferenceExperiment 574.10± 0.65 G. M. Clemence 1947

Newton’s Law 531.63±0.69 G. M. Clemence 1947??? 42.47± 0.95